How do you find the domain and range of F(x) = -2(x + 3)² - 5?

Jun 30, 2018

$x \in \mathbb{R} , y \in \left(- \infty , - 5\right]$

Explanation:

$\text{this is a polynomial of degree 2 and is defined for all }$
$\text{real values of } x$

$\text{domain is } x \in \mathbb{R}$

$\text{to find the range we require the vertex and if maximum}$
$\text{or minimum turning point}$

$\text{the equation of a parabola in "color(blue)"vertex form}$ is.

•color(white)(x)y=a(x-h)^2+k

$\text{where "(h,k)" are the coordinates of the vertex and a}$
$\text{is a multiplier}$

$f \left(x\right) = - 2 {\left(x + 3\right)}^{2} - 5 \text{ is in this form}$

$\text{with "(h,k)=(-3,-5)" and } a = - 2$

$\text{since " a <0" then maximum turning point}$

$\text{range is } y \in \left(- \infty , - 5\right]$
graph{-2(x+3)^2-5 [-28.46, 28.5, -14.22, 14.25]}